Diagrams irreducible

The key to understanding dewatering by air displacement is the capillary pressure diagram. Figure 6 presents an example typical for a fine coal suspension there is a minimum moisture content, about 12%, called irreducible saturation, which cannot be removed by air displacement at any pressure and a threshold pressure, about 13 kPa. 

Fig. 2. Some of the diagrams occurring in Mt. Articulation points are indicated by arrows, (a) and (b) are irreducible diagrams (c) and (d) are linked reducible and unlinked respectively.

The variables that appear in the single parts are therefore statistically independent. By virtue of Eq. (50) it follows that all cross products of linked reducible or unlinked parts must vanish in Mn. In consequence only the irreducible diagrams contribute to M . Furthermore, if we consider the contribution of the irreducible diagram in Figs. 2a to M3 we have the terms... 

Consider then an irreducible diagram with melon bonds in the wth semi-invariant, consisting of n lines and k nodes whose composition is k. The k nodes can be selected from the N2 defects in... 

Eq. (55) is the sum of all simple irreducible diagrams that can be formed among the k nodes, every bond representing an ffj function. (Thus every node is multiply connected. If k = 2 then B is exceptional and corresponds to the graph of two nodes and an /l2> bond.) It should be noted that flf is zero when i and j coincide (cf. Eq. (44)). 

Fig. 3. Examples of irreducible diagrams making zero contribution.

Let us now specify the nature of the dynamical irreducibility condition in Eq. (56). The conservation rides of the wave vectors (Eq. 63) impose the condition that the k of certain particles is zero in certain intermediate states. For example, in Fig. 2a particles 2 and 3 have their k zero in the second state of propagation. It may be that the structure of the diagram is such that for one or many intermediate states the k of every particle is identically zero. The diagram is then reducible (see Fig. 2c) and is not contained in Eq. (56). This leads us to extend the definition of jijru. ) so as to include in it the reducible contributions. We shall define... 

It is clear that two diagrams giving rise to the same graph may be reducible or not according to the order in which the interactions occur. For example, diagrams (a) and (b) of Fig. 7 are respectively reducible and irreducible even though one associates the same graph with them. 

D. Definitions of Young diagrams, tableaux, and operators should be understood, as well as the property of the Young operator of projecting onto an irreducible representation (Theorems 1 and 2). 

Theorem 1 The operator Y defined by (39) is, apart from a multiplicative constant, a primitive idempotent of <3n. Y operators belonging to the same Young diagram belong to the same irreducible representation, while those belonging to different diagrams belong to different representations. 

Since Qh is a direct product group, its irreducible representations are also direct products. We denote a representation of <3A by a Young diagram giving the representation of S , together with a letter g or u according as the representation is even or odd with respect to to-... 

Another correspondence between finite subgroups of SU(2) and Dynkin diagrams was given by McKay [55]. Let Rq,Ri,. .., Rn he the irreducible representations of F with Rq the trivial representation. Let Q be the 2-dimensional representation given by the inclusion F C SU(2). Let us decompose Q Rk into irreducibles, Q Rk = iCikiRh where aki is the multiplicity. Then the matrix 21 — aki)ki is an affine Cartan matrix of a simply-laced extended Dynkin diagram, An Dn Eq or Eg ... 

Formulating conditions for the energy to be stationary with respect to variations of the wavefunction P in this generalized normal ordering, one is led to the irreducible Brillouin conditions and irreducible contracted Schrodinger equations, which are conditions on the one-particle density matrix and the fe-particle cumulants k, and which differ from their traditional counterparts (even after reconstruction [4]) in being strictly separable (size consistent) and describable in terms of connected diagrams only. 

We have already discussed the relations between the four stationarity conditions. In view of their separability, the two irreducible conditions are the right choice in the spirit of a many-body theory in terms of connected diagrams. 

It is important to realize which orbital combinations are represented by the above irreducible representations. This is best achieved by looking at the diagrams in Figure 4.1 for the overlaps represented by the equations ... 

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